Assignment analyze article and answer questions about algebra

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Week 2: Embedding Algebraic Thinking throughout the Mathematics Curriculum (Vennebush, Marquez, & Larsen, 2005)

In the article, "Embedding Algebraic Thinking throughout the Mathematics Curriculum" by Vennebush, Marquez, and Larsen (2005), several activities for fostering algebraic thinking through activities targeting other content strands are offered.


a)

Compare activities you use to those presented in the article.

b) Discuss how your algebra is embedded in your school's curriculum. For example, are most students expected to complete algebra by grade eight as discussed in the article.

ARTICLE ATTACHED, PLEASE FOLLOW INSTRUCTIONS BY THE TEACHER, THIS CLASS MUST BE MASTERED. BE PROFESSIONAL.

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Embedding Algebraic Thinking throughout the Mathematics Curriculum Author(s): G. PATRICK VENNEBUSH, ELIZABETH MARQUEZ and JOSEPH LARSEN Source: Mathematics Teaching in the Middle School, Vol. 11, No. 2 (SEPTEMBER 2005), pp. 8693 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/41182860 . Accessed: 22/03/2013 23:38 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Mathematics Teaching in the Middle School. http://www.jstor.org This content downloaded from 128.192.114.19 on Fri, 22 Mar 2013 23:38:45 PM All use subject to JSTOR Terms and Conditions G. PATRICK VENNEBUSH, ELIZABETH MARQUEZ, LOVE, ALGEBRA IS WHERE YOU FIND IT. You can locateit almostanywherein the middle school curriculumif you know wheretolookandwhattolookfor.Butwith so manydemandson ourtime,we oftenforget to look.We takeproblemsat facevalue,and we assumethata geometry problemis justa geometry orthata dataanalysisactivity is onlyabout problem dataanalysis.Ifwe scratchbelowthesurface, howforalgebraic ever,we can findrichopportunities in numberexplorations, measurethinking lurking menttasks,andgeometry investigations. to Today,manyschooldistricts requirestudents beginthe formalstudyof algebrain the eighth a majorcomponentof the grade. Consequently, earlymiddleschoolyearsis dedicatedto fostering algebraicthinking, yetmiddleschoolstudentsare PATRICKVENNEBUSH,patrick.vennebush® verizon.net, formerlya developer at the Educational TestingService (ETS), is the Illuminations projectmanagerat NCTM. He is also theeditorofthe "Media Clips"columnin theMathematicsTeacherjournal.ELIZABETH is an assessment MARQUEZ, lmarquez@ets.org, developer for the National Board for ProfessionalTeachingStandardsat ETS. She wasformerly a teacherin theNorthBrunswick whereshe received SchoolDistrict,NewJersey, AwardforExcellencein Math thePresidential and Science Teaching. JOSEPH LARSEN, at jplar@aol.com, is an educationalconsultant ETS whohelpedtodevelopand nowassessesthe a mathematics middleschoolPraxisexams.He wasformerly departmentchair at Council Rock School District in Pennsylvania.This articleis theresultof theirworkwith teachersacrossthecountry in developing TeacherAssistance Packagesforalgebraand prealgebrainstruction. 86 and JOSEPH LARSEN alsoexpectedtolearnmanyconceptsandskillsthat are not directly relatedto algebra.Accordingto Principlesand Standardsfor School Mathematics shouldreceive (NCTM2000),theAlgebraStandard about one-third of the emphasisin the middle and grades. (A graphicon page 30 of Principles Standards"showsroughlyhow the ContentStandardsmightreceivedifferent emphasesacrossthe gradebands."This graphicis notmeantto dictate the specificnumberof lessons for each topic theyear,butitdoes suggestthattheAlthroughout 30 gebra Standardshouldreceiveapproximately percentoftheemphasisinthemiddlegrades.) Withso littletimereservedfortheAlgebraStandardintheearlymiddleschoolyears,howis itpossible to fosteralgebraicthinking whilealso sufficientlyaddressingthe Standardsof Numberand Measurement,and Data Operations,Geometry, andProbability? Analysis One effective approachis to use richtasksthat addressmorethanone Standard.Principlesand do "not Standards statesthattheContent Standards curriculum neatlyseparatetheschoolmathematics intononintersecting subsets.Becausemathematics the areas as a disciplineis highlyinterconnected, describedby the Standardsoverlapand are integrated"(NCTM2000,p. 30). Teachers oftenchoose rich activitiesforthe oftopics classroomthatallowforthe exploration whenthey frommorethanone standard. However, theconnections betweentheStanfailto highlight topromote dards,teachersmaylose an opportunity algebraicskillsin the middlegrades.This article identifies someofthealgebraimplicit within activitiesthatemphasizeotherconcepts,connectstraditionalalgebraproblems totheotherfourStandards, andoffers formodifying so that activities strategies theywillbe abletofoster algebraicthinking. MATHEMATICS TEACHING IN THE MIDDLE SCHOOL This content downloaded from 128.192.114.19 on Fri, 22 Mar 2013 23:38:45 PM All use subject to JSTOR Terms and Conditions WhatIs Algebraic Thinking? AND MANY MENTIONTHE WORD "ALGEBRA" - thinkof teachers included students and people variablesacx's and equationsand manipulating a of rules. to set manipAlthough symbolic cording one ofthemostimportant ulationis arguably parts ofalgebra,itis nottheonlypart.Algebraic thinking muchmore. involves Algebrais oftendescribedas thestudyofgenerInsteadof dealingsolelywith alized arithmetic. itfocuseson operations numberand computation, and processes.Greenesand Findellsuggestthat involve thebigideasofalgebraicthinking represenof balance, tation, meaning reasoning, proportional and and inductive and functions, variable,patterns deductivereasoning(Greenesand Findell1998). contendsthatalgebraicthinking Kriegler Similarly, combinestwo veryimportant components: algeand funcbraicideas,including variables, patterns, tools,specifically tions,withmathematical thinking (e.g., diagrams, problemsolving,representation words,tables),and reasoning(Kriegler2004). In that is usedinanyactivity short,algebraicthinking of the with one combinesa mathematical process such as in ideas understanding patalgebra, big with situations ternsand functions, representing models,and analyzsymbols,usingmathematical foralgebraic As such, opportunities ing change. as illustrated in arise contexts, by many thinking theexamplesthatfollow. ina Number Thinking Algebraic Sense Task SYSTEMIMPLICNUMERATION THEPLACE-VALUE of the basic some conceptsofalgeitlyincorporates number of and the bra, rely operations algorithms heavilyon the"lawsofalgebra"(NationalResearch Council2001,p. 256).Itis notsurprising, then,that On each turn,a playerspinsthetwospinnersshownbelow. The scorefortheturnis foundbytakingtheproductofthe numberson whichthespinnersstop.Forexample,ifthefirst spinnerstopson -5 and the secondspinnerstopson 2, the scoreforthatturnwouldbe -5 x 2,or-10. totalis keptbyaddingthescoreforeach turn,and A running theplayerwiththehighesttotalafter10turnsis thewinner. Fig. 1 The rules forthe "positivelynegative" game activities dealingwithnumber manymiddle-grades even if and operationsinvolvealgebraicthinking, varithetaskdoes notspecifically targetpatterns, ables,orotheralgebraicideas. thatcombinesnumber One classroomactivity is the"positively senseandalgebraicthinking negawithIntegers tive"game,takenfromOperating (ETS 2003c);therulesforthegamearegiveninfigure1. involvesthemultiplication This stimulating activity ofpositiveandnegative andaddition integers; open andthistopicusuan introductory algebratextbook fewpages.As students play allyappearson thefirst unawarethattheyarepracthisgame,theyareoften ticingbasic operationsand developingalgebraic students' skillsin ideas.The tasksubtlyimproves mathematics. is an exampleofa questionthat The following to genIt asks students fostersalgebraicthinking. eratea sequenceofscoresforthe"positively negaoutcome. tive"gamethatwouldleadtoa particular VOL. 11, NO. 2 • SEPTEMBER 2005 87 This content downloaded from 128.192.114.19 on Fri, 22 Mar 2013 23:38:45 PM All use subject to JSTOR Terms and Conditions After Jacoband Simonehad each taken7 turns, 63to12.Simonecouldnotbelieve Jacobwasleading Showa sequence thatshehadscoredonly12points! ofscoresthatwouldyielda totalscoreof12points from ETS 2003c,p. 11) 7 turns. after (Adapted becauseit Thisquestioninvokesalgebraicthinking, combinestheprocessofproblemsolvingwiththe to It also forcesstudents algebraicidea ofpatterns. thinkaboutissuesofbalance- one ofthebigideas - byconsidering addilistedbyGreenesandFindell tiveinverses,and howpositivevaluescan be canvalues. celedbynegative When the "positivelynegative"spinnersare as shownin spun,thereareninepossibleproducts, thegridbelow: 6 4-5 14-56 2 8 -10 12 -3 -12 15 -18 Task and a Geometry Thinking Algebraic THE DECREASINGVOLUME TASK,TAKEN FROM Volumeand Dimension(ETS 2003b),iniExploring whichdimension tiallyasks studentsto determine ofa5x7xl5 boxshouldbe decreasedby1 unitto producethegreatestdecreaseinvolume.Students are thenasked to generalizewhichdimensionof box shouldbe reducedby1 unitto anyrectangular decreaseinvolume. yieldthegreatest For the specificcase of a 5 unitx 7 unitx 15 unit box, the typicalsolutionstrategyinvolves studentsproceedas testingeach case. Generally, follows: • Ifthe5-unit sideis reduced,thenewboxwillbe 4x7x15, andhavea volumeof420cubicunits. • Ifthe7-unit sideis reduced,thenewboxwillbe 5x6x15, andhavea volumeof450cubicunits. • Ifthe15-unit sideis reduced,thenewboxwillbe 5 x 7 x 14,andhavea volumeof490cubicunits. studentsconcludedthat Fromtheseobservations, desidewouldyieldthegreatest the5-unit reducing creaseinvolume. Studentsmightnoticethateach numberin the bottomrowis equal to theoppositeofthesumof in thefirst thetwonumbersaboveit;forinstance, column,-12 is the oppositeof4 + 8. Recognizing this patternis usefulin answeringthe question aboutJacoband Simone,becausethreeturnswith Twosetsof scoresof4,8,and-12 resultin0 points. scoreof thesethreescoresas wellas an additional 12pointswillyieldthedesiredresult: 4 + g + (-12) + 4 + 8 + (-12) + 12= 12 othersetsofthreescoreswillresultin Similarly, 0 points:(-5, -10, 15), (6, 12,-18), (4, 6, -10), and withan (6,6,-12). Anytwoofthesesets,combined willyielda totalscoreof turnof12points, additional workatthebot12pointsafter7 turns.The student tomoffigure2 uses thisideaofbalancetoobtaina correctsequenceforSimone'sspins. theidea ofsymbolic In thecontext manipulation, tothe tocorrespond ofbalanceis generally thought rule"Anything youdo toone sideofthe procedural anditis imporequation, youmustdo totheother," thisrulewhendetantforstudentsto understand a broader withsymbols.However, velopingfluency tounderstanding viewofbalancerefers equivalence and opposites.Althoughthe "positively negative" with to students proficiency acquire helps game it enables them also and numbers, negative positive ofbalance. todevelopan intuitive understanding 88 Fig. 2 A studentsolutionforthe Simone and Jacob problem using the idea of balance MATHEMATICS TEACHING IN THE MIDDLE SCHOOL This content downloaded from 128.192.114.19 on Fri, 22 Mar 2013 23:38:45 PM All use subject to JSTOR Terms and Conditions Based on theexploration ofa5x7xl5 box,the Volume task Decreasing mayappearto be solely even the Indeed, geometric. generalcase regarding a box ofanysize can be solvedwithgeometric reawhoseworkis soning,as was donebythestudent showninfigure3. However, thetaskextendstoalbecausestudents areaskedtogengebraicthinking eralizetheirfindings. The student workinfigure4 does notuse algebraicsymbols, butitdoes use inductivereasoningto reacha conclusion. Usingthe previousresultsfromthe5 x 7 x 15 box,as wellas boxeswithdimensions of3 x 4 x 5 and considering 5x2x8, the studentrecognizesa patternfrom threespecificexamples.Moretothepoint,thestudent'sworkinfigure5 exhibitsobviousalgebraic inthatthestudent usedvariablesto reprethought sentandanalyzethesituation. The student pictured thethreeslicesthatcouldbe removed fromthebox andrepresented theirvolumeswithformulas using theslice /,wyand/г.If/гis theshortest dimension, withdimensions w x I xl has thegreatest volume. thestudent concludedthattheboxwith Therefore, volumeIwh- wll has the greatestdecreasefrom theoriginal. Thisis a generalized formofthestrategyused infigure4. Sincealgebrais "thegeneralizationofarithmetic," thissolutionclearlydemonstratesa greatdealofalgebraicthinking. The conclusion infigure4 is based on boxesof severalsizes,whereastheconclusion infigure5 is based on everypossiblebox size,Ixwxh. Consequently,even thougheach studentreceivesfull creditforhis orherwork,theexplanation givenby thestudent infigure5 is morecomplete. Showing students an algebraicapproachto solvingproblems like thismaybe used to introducethe powerof In addition, thisproblem symbolicrepresentation. canbe revisited as students acquirealgebraicskills andas theyencounter othersituations aboutwhich theyareaskedtogeneralize. Fig. 3 A geometricsolutionto the Decreasing Volumetask Fig. 4 An inductivereasoning solutionto the Decreasing Volumetask inan AlgebraTask Data Analysis SOMETIMESTHE CONNECTION BETWEENALGEbraandotherStandards turnsup whereyouwould least expectit.The Odd Integersproblembelow was givento a groupofstudents to determine their totranslate intosymbolsand wordproblems ability tosolveequations. The sumoffourconsecutive oddintegers is 112. What is the greatestof these fourintegers? (MATHCOUNTS2003,p. 51) Oneteacherwhousedthisproblem expectedstudentstorepresent eachofthefouroddnumbers with an algebraic setupan equation, solvefor expression, Fig. 5 An algebraic solutionto the Decreasing Volumetask andinterpret theresults. a variable, Figure6 shows thatusedthisexpectedmethod, onesolution as well as twoothersuccessful solutions. SolutionA exhibitsa strategy typicalofwhatwe and mightexpectfroma first-year algebrastudent, solutionВ employsa guess-and-check Alstrategy. solution В reliesentirely on thoughequallycorrect, numbersense and computation. SolutionC, aldemonstrates numbersenseprincithoughatypical, amountofalgebraicthinking. plesanda significant VOL. 11, NO. 2 ■ SEPTEMBER 2005 89 This content downloaded from 128.192.114.19 on Fri, 22 Mar 2013 23:38:45 PM All use subject to JSTOR Terms and Conditions Fig. 7 A representationshowingthe average of fourconsecutiveodd integers Fig. 6 Solutions to the Odd Integersproblem The firststep (dividing by 4) findstheaverageof In stating thatthe"third number thefournumbers. is 28 + 1,"thestudentseemsto use a variableiman unknown theaverto represent plicitly quantity: age ofthefournumbersis x,andthethirdnumber is x + 1,usingthesymmetry ofodd numberson eithersideofthemean. The idea ofaveragingis oftendescribedas an "eveningout"of data: take a littlefromthispile to averages,soandadd ittothatpile.Byreferring lutionС connectsalgebrato dataanalysis.Each of theintegerscan be represented byan expression, 90 an arrangement ofalgebratiles,or a diagram.As figure7 shows,thealgebraicaverageofthefour integersis x + 3: removea squarefromx + 4 and x+ 6 add ittox + 2, andremovethreesquaresfrom andadd themtox. Numerically, theaverageofthe 4 = 28.As a result, x + 3 = 28, fournumbersis 112-s= whichmeansthatx 25, and the greatestofthe fourintegersis x + 6, or31. The Odd Integersproblemis a valuabletaskfor middleschool students,since it invitessolutions thatinvolvemultiple strandsand enablesteachers to highlight how one problemcan be solvedwith variousstrategies. Unlikethe"positively negative" game,whichis a numbersense taskthatincorporates algebraicideas, and DecreasingVolume, whichintegrates algebraicconceptsintoa geometan rictask,theOdd Integersproblemis foremost - onethatmightappearina typical algebraproblem algebracourse- thatallowsfortheexeighth-grade ofa topicfromdataanalysis. ploration Tasksto Promote Modifying Thinking Algebraic ANOTHER EXAMPLE OF THE CONNECTION BEtweendata analysisand algebra can be seen in the Bookworms taskfrom Data and Making Analyzing Predictions (ETS 2003a).As showninfigure8, the taskpresentsdatacollectedfromstudentsurveys. mostofthequestionsassess a student's Although topredataanalysisskills,question4 asks students MATHEMATICS TEACHING IN THE MIDDLE SCHOOL This content downloaded from 128.192.114.19 on Fri, 22 Mar 2013 23:38:45 PM All use subject to JSTOR Terms and Conditions dietresultsfortheentireschoolpopulation. A solutionthatemploysalgebraicthinking forquestion4 oftheBookworms taskis shownin figure9. The methodused infigure10, althoughnotstrictly algebraic,also requiresthatstudentsapplyproportionalreasoning,whichis neededformanyalgebraictasks. infigures9 and 10 are bothcorThe solutions to showingstudents rect,andthereare advantages thevalueofeach method.On theone hand,bynot a particular solutionstrategy, thistaskis requiring valuableforexpandingstudents'problem-solving skills.Ontheotherhand,theproblem canbe usedto assess a student's totranslate wordproblems ability to equationsand to manipulate To obtain symbols. evidenceforthispurpose, itmaybe necessary toask a follow-up suchas thefollowing: question, Explainhowtheproblemmightbe solvedusing a proportion. This minormodification makestheBookworms taskappropriate forassessingproportional reasonHowing,whichis essentialforalgebraicthinking. ever,substantial changesare occasionally required to makea tasksuitableforstudents. For example, the finalpartof the DecreasingVolumetask reto explainthesolutionforanysize quiresstudents forstudents box,butsuch a taskmaybe difficult who have minimalexperiencein generalizing results.Addingthe information shownin table 1 shouldhelp studentsidentify a patternas dimensionsare changed.Usingtheresultsfromtable 1, the studentcan be askedto explainwhichdimension shouldbe reducedby 1 unitto producethe decreaseinvolumeforanysizebox. greatest As shownbythelastrowofthetable,themodificationcanincludevariables, forthediscovallowing and assessmentof symbolic ery,reinforcement, andmanipulation skills. representation as showninfigure6, theOdd Integers Similarly, canbe solvedusingalgebra, number problem sense, ordataanalysis To ensurethatthetaskretechniques. toapplyalgebraic quiresstudents thinking, anyofthe additions ormodifications couldbe used: following • Ifn is an oddinteger, whatexpressions couldbe usedtorepresent thenextthreeoddnumbers? • Fourconsecutive odd integersare represented byn,n + 2, n + 4,andn + 6,andthesumofthese fournumbersis 112.Solvetheequationn + in + theval2) + (n + 4) + (n + 6) = 112to determine ues ofthefourintegers. • The smallestoffourconsecutive odd integers is n,andthesumofthesefournumbersis 112.In the threeblanksbelow,writeexpressionsthat дО jÖ' 'j-+^t ^^^ f*^^^^^] '^ J ^^^^ '^^^^^ BOOKWORMS i tooka surveyoftheirclassmatesto determine Juananc*Britney whattypeofbooks theymostenjoyreading.Juansurveyedthe boys,and Britney surveyedthegirls.The resultsare shownin thetablebelow. TypesofBooksThatStudentsMostLiketo Read Type of Book Action Juan (Boys) 8 Britney(Girls) 18 Mystery Science Fiction 8 13 18 18 Nonfiction 6 11 Total Total Q 1. Completethetableabove. 2. On a separatesheetofpaper,makean appropriategraphicalrepresentation ofthe data collectedbyJuan. 3. What percentof thegirlssurveyedbyBritneylikeactionbooks best? Show how you foundyouranswer. I (Л cr
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Running Head: ALGEBRAIC THINKING OUTLINE

ALGEBRAIC THINKING OUTLINE
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ALGEBRAIC THINKING OUTLINE

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ALGEBRAIC THINKING OUTLINE

Thesis statement: There are many instances that invoke the use of algebraic reasoning in real
life. Almost every situation can be solved using an algebraic expression or algebraic. When
solving various problems in real life, you need to come up with generalized algebraic equations
relating to each part of the problem.

Answer

ALGEBRAIC THINKING OUTLINE

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Running Head: ALGEBRAIC THINKING
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I was struggling with this subject, and this helped me a ton!

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